Snells law

Snell’s law—also known as the Snell–Descartes law, the Ibn Sahl law, or the law of refraction—is a fundamental relationship in optics describing how light or other waves change direction when passing between two isotropic media of different refractive indices. It quantifies the bending of a ray at an interface by relating the angle at which the wave strikes the boundary to the angle at which it emerges in the second medium. In practical optics the law is essential for ray tracing, interpreting refractive behaviour in lenses and prisms, and determining the refractive index of materials.
For two media with refractive indices n1n_1n1​ and n2n_2n2​, and angles of incidence and refraction θ1\theta_1θ1​ and θ2\theta_2θ2​ measured with respect to the normal, the law states that
sin⁡θ1sin⁡θ2=n2n1=v1v2,\frac{\sin \theta_1}{\sin \theta_2}=\frac{n_2}{n_1}=\frac{v_1}{v_2},sinθ2​sinθ1​​=n1​n2​​=v2​v1​​,
expressing equivalence between refractive index ratios, wave-speed ratios, and the corresponding angular relationship. When a wave slows on entering a higher-index medium, the refracted ray bends towards the normal; when it enters a lower-index medium it bends away. These principles hold for light in media such as air, water, glass, and also in engineered metamaterials exhibiting negative refractive indices, where the ray may refract at a negative angle.

Historical development

The first accurate formulation of the law originated in 984 CE with the Persian mathematician Ibn Sahl, working at the Abbasid court in Baghdad. In his treatise On Burning Mirrors and Lenses, he derived the relationship governing light refraction and used it to design lens shapes free from spherical aberration. Although Ptolemy earlier attempted to formulate an empirical law of refraction, inaccuracies in his recorded data limited its validity, and later analysis revealed adjustment of observations to match expectations.
The law was independently rediscovered several times. Thomas Harriot obtained it in 1602 but never published his findings. In 1621 the Dutch astronomer Willebrord Snellius derived the modern trigonometric form, which remained unpublished during his lifetime. René Descartes later obtained an equivalent formulation and presented it in Dioptrique (1637), deriving the result via a momentum analogy. Although contemporaries suspected Descartes of borrowing Snell’s work, later scholarship has found no supporting evidence. Pierre de Fermat derived the law from his principle of least time, demonstrating the connection between refraction and variational ideas. His treatment relied on the assumption that light moves more slowly in denser media, a view supported experimentally by later work.
Christiaan Huygens provided an elegant wave-based derivation in his Traité de la lumière (1678), using secondary wavefronts to reconstruct refracted propagation. With the emergence of electromagnetic theory in the nineteenth century and the development of Maxwell’s equations, Snell’s law acquired a firm theoretical basis. More recent advances have generalised the law to nonlinear optical boundaries and to engineered metasurfaces that can dynamically steer reflected or refracted beams.

Explanation and physical interpretation

Refractive index expresses the factor by which light slows relative to its speed in vacuum. Snell’s law determines how a ray changes direction when entering a medium of different index:

• If n2>n1n_2>n_1n2​>n1​, light slows down and bends towards the normal (θ2<θ1\theta_2<\theta_1θ2​<θ1​).
• If n2<n1n_2<n_1n2​<n1​, light speeds up and bends away from the normal (θ2>θ1\theta_2>\theta_1θ2​>θ1​).
• The process is reversible: if the ray retraces its path from medium 2 to medium 1, the same angular relation holds.

In anisotropic materials such as birefringent crystals, a single incident ray may split into ordinary and extraordinary rays. The ordinary ray adheres to Snell’s law; the extraordinary ray may not lie in the plane of incidence and follows a different rule determined by the crystal’s optical axis.
For monochromatic light, the law can also be written using wavelengths:
sin⁡θ1sin⁡θ2=λ1λ2,\frac{\sin\theta_1}{\sin\theta_2}=\frac{\lambda_1}{\lambda_2},sinθ2​sinθ1​​=λ2​λ1​​,
reflecting the proportional shortening of wavelength as light enters a higher-index medium.

Derivations

Snell’s law can be derived from several foundational principles:

• Fermat’s principle: Light follows the path requiring the least time. Computing the optical path length and setting its derivative to zero yields the standard law.
• Wavefront geometry: Huygens’ principle shows that the geometry of secondary wavelets at an interface reproduces the law naturally.
• Phase matching: Maxwell’s equations require that the tangential component of the wave vector remains continuous across the interface, giving rise to Snell’s relationship.

The classic analogy equates the high-index medium to a region where movement is slower, as in a rescuer crossing from sand into water; the quickest path to a target requires angled entry, mirroring the refracted path of a light ray.

Applications

Snell’s law underpins much of geometrical and applied optics. It is fundamental to:

• lens and prism design,
• optical instrument calibration,
• determining refractive indices through controlled-angle measurements,
• modelling atmospheric refraction,
• interpreting mirages and other refractive phenomena, and
• engineering metamaterials that achieve unconventional refraction patterns.